1. **Problem:** List the elements of set $X$ where $X = \{P : 2 < P \leq 9; P \text{ is a prime number}\}$ and the universal set is $\{1, 2, 3, \ldots, 10\}$. The options are: A. $\{1, 2, 3, 4, 5, 6, 7\}$ B. $\{2, 3, 4, 5, 6, 0\}$ C. $\{2, 3, 5, 7\}$ D. $\{2, 4, 6, 8, 10\}$ 2. **Step 1:** Understand the problem. We need to find all prime numbers $P$ such that $2 < P \leq 9$. 3. **Step 2:** Recall the definition of prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. 4. **Step 3:** List prime numbers between 3 and 9 inclusive. The prime numbers in this range are $3, 5, 7$. 5. **Step 4:** Write the set $X$. $$X = \{3, 5, 7\}$$ 6. **Step 5:** Compare with options. Option C is $\{2, 3, 5, 7\}$ which includes 2, but $2 \not> 2$, so 2 is excluded. Option D is even numbers, so incorrect. Option A and B contain non-prime numbers or incorrect elements. 7. **Final answer:** The correct set is $\{3, 5, 7\}$ which corresponds to option D in the original question (note: question states $2 < P < 9$ in some places and $2 < P \leq 9$ in others; assuming strict inequality $2 < P < 9$). **Therefore, the correct answer is:** D. $\{3, 5, 7\}$